Editing AxisymmetricConfigurations/SolutionStrategies (section)

==Lagrangian versus Eulerian Representation==
In our overarching specification of the set of [[PGE#Principal_Governing_Equations|''Principal Governing Equations'']], we have included a,

<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the Euler Equation,

{{Math/EQ_Euler01}}

[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.55)
</div>

When seeking a solution to the set of governing equations that describes a steady-state equilibrium configuration &#8212; as has already been suggested in our [[PGE/Euler#Eulerian_Representation|accompanying discussion of "other forms of the Euler equation"]] &#8212; it is preferable to start from an,
<div align="center">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

{{Math/EQ_Euler02}}
</div>


because steady-state configurations are identified by setting the ''partial'' time derivative, rather than the ''total'' time derivative, to zero.  Notice that if the objective is to find an equilibrium configuration in which the fluid velocity is not zero &#8212; consider, for example, a configuration that is rotating &#8212; then throughout the configuration, the velocity field must be taken into account, in addition to the gradient in the gravitational potential, when determining the pressure distribution.  Specifically, for steady-state flows, the required relationship is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{1}{\rho} \nabla P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla \Phi - (\vec{v} \cdot \nabla) \vec{v} \, .</math>
  </td>
</tr>
</table>

As we also have [[PGE/Euler#in_terms_of_the_vorticity:|mentioned elsewhere]], by drawing upon a relevant [https://en.wikipedia.org/wiki/Vector_calculus_identities#Dot_product_rule dot product rule vector identity], this expression can be rewritten in terms of the fluid vorticity, <math>~\vec\zeta \equiv \nabla\times\vec{v}</math>, as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{1}{\rho} \nabla P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla \biggl[ \Phi + \frac{1}{2}\vec{v}\cdot \vec{v} \biggr] -  \vec\zeta \times \vec{v} \, .</math>
  </td>
</tr>
</table>

<span id="CentrifugalPotential">In certain astrophysically relevant situations &#8212; such as the adoption of any one of the ''simple rotation'' profiles identified immediately below &#8212; the nonlinear velocity term involving the "convective operator" can be rewritten in terms of the gradient of a scalar (centrifugal) potential, that is,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\nabla \Psi \, .</math>
  </td>
</tr>
</table>
In such cases, the condition required to obtain a steady-state equilibrium configuration is given by the considerably simpler mathematical relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{1}{\rho} \nabla P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla \biggl[ \Phi + \Psi \biggr]  \, .</math>
  </td>
</tr>
</table>

In the subsection of this chapter (below) titled, [[#Double_Check_Vector_Identities|''Double Check Vector Identities,'']] we explicitly demonstrate for four separate "simple rotation profiles"  that these three separate steady-state balance expressions do indeed generate identical mathematical relations.